Keywords: convection-diffusion equation, streamline upwind Petrov Galerkin, proper generalized decomposition, separated representations, finite sum decomposition.

The main aim of this paper is to present one application of a new numerical strategy able to circumvent some of the numerical difficulties due to some convection-diffusion equations. It is well-known that the behaviour of standard finite elements in the resolution of convection-diffusion (-reaction) equations is not good. The use of numerical stabilization terms in the differential equation has been developed and discussed since some years ago. The use of streamline upwind Petrov Galerkin (SUPG) stabilization technique offers exact solutions in one-dimensional problems, but does not work well in two- or three-dimensional spaces [1]. The use of the proper generalized decompositions strategies [2,3] composed by a finite sum of one-dimensional products provides us the possibility of developing this technique applying SUPG stabilization in the resolution of the convection-diffusion problems defined in multidimensional spaces. Among the very numerous methods that have been proposed for the stabilization of convection-diffusion equations, the SUPG method is one of the most extended. The major drawback of this method is that it introduces some cross-wind artificial diffusion when applied to problems defined in N dimensional spaces, N>1. The use of separated representations allows for an easy treatment of problems defined in spaces of a high number of dimensions. Here, the use of finite sums of separable functions will be used towards an approximation to a sequence of different one-dimensional problems, for which exact SUPG stabilization exists. In this work we describe the framework of the class of convection-diffusion (-reaction) problems we deal with. We make a brief description of the PGD method used to apply the stabilization technique and we describe some examples in order to show the behaviour of the technique in the simulation of this class of problem.

1

J. Donea, A. Huerta, "Finite Element Methods for Flow Problems", John Wiley and sons, 2003.

2

A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, "A New Family of Solvers for Some Classes of Multidimensional Partial Differential Equations Encountered in Kinetic Theory Modeling of Complex Fluids", Journal of Non-Newtonian Fluid Mechanics, 139(3), 153-176, 2006. doi:10.1016/j.jnnfm.2006.07.007

3

A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, "A New Family of Solvers for Some Classes of Multidimensional Partial Differential Equations Encountered in Kinetic Theory Modeling of Complex Fluids. Part II: transient simulation using space-time separated representations", Journal of Non-Newtonian Fluid Mechanics, 144, 98-121, 2007. doi:10.1016/j.jnnfm.2007.03.009

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